Ding Haojiang

General solutions for coupled equations for piezoelectric media

Abstract Three general solutions are obtained for the coupled dynamic equations for a transversely isotropic piezoelectric medium. These solutions are expressed in terms of the two functions ψ and F, where ψ satisfies a second-degree partial differential equation and F a sixth-degree partial differential equation, respectively. If the terms concerning the derivatives of time are removed, the results become three general solutions for the corresponding equilibrium equations, in which the function F can be represented by functions Fi (i = 1, 2, 3), each of which satisfies a second-degree partial differential equation by utilizing a generalized Almansi theorem; and the solution Wang and Zheng [Int. J. Solids Structures32, 105–115 (1995)] obtained is proved to be consistent with one case of one of the three general solutions. When the constants e15 = e31 = e33 = 0 the piezo-electric coupling is absent; then, two of the solutions reduce to the elasticity general solutions for a transversely isotropie medium, one of which is the result Hu [Acta Scientia Sin.2(2), 145–151 (1953)] obtained; the other one has not been published. Last, the solution in the limiting explicit form for the problem for a half-space with concentrated loads at the boundary is obtained by utilizing the general solutions.

On the Green's functions for two-phase transversely isotropic piezoelectric media

Abstract This paper is concerned with the problem of point force and point charge applied in the interior of an infinite two-phase transversely isotropic piezoelectric solid. Based on the general solutions, by using the method of the image source, a series of displacement functions are constructed. The Greens functions are obtained when arbitrary constants are determined by the boundary conditions on the interface. Furthermore, we reduce the present solutions to the extension of Mindlin results and of Lorentz results for semi-infinite transversely isotropic piezoelectric materials by suitable substitutions of boundary conditions on the interface.

A boundary integral formulation and 2D fundamental solutions for piezoelectric media

Abstract First, based on the basic equations of piezoelectricity, the boundary integral formulation is derived by utilizing the reciprocal-work theorem. Second, for the plane problem of piezoelectric media, one general solution in terms of ‘harmonic functions’ is derived and the fundamental solutions are therefore obtained. Numerical calculations by boundary element method (BEM) are performed to give the stress concentration coefficient of an infinite piezoelectric plane with a circular hole as well as the stress and electric intensity factors of a central crack locating on an infinite plate. Results are found to agree well with the exact solutions.

The elliptical Hertzian contact of transversely isotropic magnetoelectroelastic bodies

First, the general solution for transversely isotropic magnetoelectroelastic media is given concisely in form of five harmonic functions. Second, the extended Boussinesq and Cerruti solutions for the magnetoelectroelastic half-space are obtained in terms of elementary functions by utilizing this general solution. Third, the coupled fields for elliptical Hertzian contact of magnetoelectroelastic bodies are solved in smooth and frictional cases. At last, the graphic results are presented.

Free axisymmetric vibration of transversely isotropic piezoelectric circular plates

Abstract Based on three-dimensional elastic theory of piezoelectric materials, the axisymmetric state space formulation of piezoelectric laminated circular plates is derived. Finite Hankel transforms are used and the boundary variables in free terms are replaced, for two kinds of boundary conditions, to obtain ordinary differential equations with constant coefficients. Regarding the axisymmetric free vibration problem, two exact solutions for two different boundary conditions are found. Discarding piezoelectric effect, the exact solutions for transversely isotropic circular laminates are also obtained through the same procedure. Numerical examples are given and compared with those of Finite Element Method (FEM) .

Nonaxisymmetric free vibrations of a spherically isotropic spherical shell embedded in an elastic medium

Based on three-dimensional elastic theory, the nonaxisymmetric free vibrations of a spherically isotropic spherical shell embedded in an elastic medium are studied in the paper. Three displacement functions are introduced to simplify the governing equations of a spherically isotropic medium for free vibrational problem. The Pasternaks assumption is adopted for the elastic medium, for which the P-ζ relation in the spherical coordinates is derived by the principle of minimum potential energy. It is found that the vibration of an embedded spherical shell can be divided into two classes, as the case in vacuum. The first class is identical to the corresponding one in vacuum, and the second has changed due to the effect of the surrounding medium. Numerical results are carried out to clarify the effect of relative parameters.

Three-dimensional analysis of a thick FGM rectangular plate in thermal environment

The thermal behavior of a thick transversely isotropic FGM rectangular plate was investigated within the scope of three-dimensional elasticity. Noticing many FGMs may have temperature-dependent properties, the material constants were further considered as functions of temperature. A solution method based on state-space formulations with a laminate approximate model was proposed. For a thin plate, the method was clarified by comparison with the thin plate theory. The influences of material inhomogeneity and temperature-dependent characteristics were finally discussed, through numerical examples.

Solutions to equations of vibrations of spherical and cylindrical shells

The governing equations of the free vibrations of spherical and cylindrical shells with a regular singularity, are solved by Frobenius Series Method in the form of matrix. Considering the relationship of the roots of the indicial equation, we get some various expressions of solutions according to different cases. This work lays a foundation of solving certain elastic problems by analytical method.

General solution of plane problem of piezoelectric media expressed by “harmonic functions”

First, based on the basic equations of two-dimensional piezoelectroelasticity, a displacement function is introduced and the general solution is then derived. Utilizing the generalized Almansis theorem, the general solution is so simplified that all physical quantities can be expressed by three “harmonic functions”. Second, solutions of problems of a wedge loaded by point forces and point charge at the apex are also obtained in the paper. These solutions can be degenerated to those of problems of point forces and point charge acting on the line boundary of a piezoelectric half-plane.

Solutions for transversely isotropic piezoelectric infinite body, semi-infinite body and bimaterial infinite body subjected to uniform ring loading and charge

Abstract Based on the fundamental solutions for transversely isotropic piezoelectric materials, the fundamental solutions of axisymmetric problems are derived by integration and explicit expressions for three possible cases of different characteristic roots and multiple roots are all presented. In the case of s 1 ≠ s 2 ≠ s 3 ≠ s 1 , based on the Greens functions for semi-infinite piezoelectric body and bimaterial infinite piezoelectric body, the Greens functions for axisymmetric problems of semi-infinite body and bimaterial infinite body are obtained. Taking PZT-4 as an example, numerical computations are conducted by use of the fundamental solutions to axisymmetric problems. Comparison of the calculated results with those of FEM shows good agreement between them.